How do you do this There are 123 chocolates in a box They are either plain milk or white The ratio of plain to milk is 2 to 3 The ratio of milk to white is 7 to 2 Find the number of plain chocolates?

Answer 1

As "milk" appears in both ratios you need to create equivalent ratios with the same value of "milk" in each; then you can combine the two ratios in to one ratio for all three and use that ratio to solve the problem.

Equivalent ratios are like equivalent factions: whatever you multiply/divide one side of the ratio by, you must multiply/divide the other side by the same amount and you will have an equivalent ratio - this is how ratios are simplified.

The first ratio plain : milk = 2 : 3

The second ratio milk : white = 7 : 2

To avoid fractions, use the lcm of the value of the "milk" in both ratios: ie the lcm(3, 7) = 21 as the value of "milk" in the equivalent ratios - divide the lcm by the current value to find what number you need to multiply the ratio by:

The first ratio can be multiplied by 21 ÷ 3 = 7 to give:

plain : milk = 2 : 3 = 2×7 : 3×7 = 14 : 21

The second ratio can be multiplied by 21 ÷ 7 = 3 to give:

milk : white = 7 : 2 = 7×3 : 2×3 = 21 : 6

The two ratios can now be combined in a single ratio with three terms as the "milk" is 21 in each case, forming:

plain : milk : white = 14 : 21 : 6

From this we can find how many of each type of chocolate there are, and in particular how many plain ones there are:

There are 14 + 21 + 6 = 41 parts

→ each part is worth 123 chocs ÷ 41 parts = 3 chocs per part

→ There are 14 parts × 3 chocs per part plain = 42 plain chocolates

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To check we can also work out the quantity of the other two types of chocolates:

& There are 21 parts × 3 chocs per part milk = 63 milk

& There are 6 parts × 3 chocs per part white = 18 white

This gives a total box of 42 plain + 63 milk + 18 white = 123 chocolates

And recalculate the ratios, simplifying them:

Ratio of plain : milk = 42 : 63 = 2×21 : 3×32 = 2 : 3

Ratio of milk : white = 63 : 18 = 7×9 : 2×9 = 7 : 2